The numerator of a positive fraction is 2 less than its denominator. By adding 1½ to the fraction, the fraction is reversed. Find the original fraction.
Help me please
[(x-2) / x] + 3/2 = x / (x-2)
2(x-2)^2 + 3x^2 - 6x = 2x^2
2x^2 - 8x + 8 + 3x^2 - 6x = 2x^2 ... 3x^2 - 14x + 8 = 0
there is one positive solution ... 2/4 + 3/2 = 4/2
To find the original fraction, let's first assign variables to the numerator and denominator.
Let "x" be the numerator and "y" be the denominator.
According to the problem, the numerator is 2 less than the denominator, so we can write the equation:
x = y - 2
Now, let's add 1½ to the fraction and reverse it:
(x + 1½) / (y + 1½) = (y + 1½) / (x + 1½)
To solve this equation, we can cross multiply:
(x + 1½) * (x + 1½) = (y + 1½) * (y + 1½)
Expanding both sides:
(x^2 + 3x + 9/4) = (y^2 + 3y + 9/4)
Now, let's substitute x = y - 2 in this equation:
(y - 2)^2 + 3(y - 2) + 9/4 = y^2 + 3y + 9/4
Expanding and simplifying:
y^2 - 4y + 4 + 3y - 6 + 9/4 = y^2 + 3y + 9/4
Now, let's combine like terms:
y^2 - y^2 - 4y + 3y + 4 - 6 + 9/4 - 9/4 = 0
Simplifying further gives:
-1y + 1/2 = 0
Let's isolate "y" by moving the constant term to the other side:
-1y = -1/2
Dividing by -1 on both sides:
y = 1/2
Since the denominator, y, is 1/2, we can substitute this value into the equation x = y - 2:
x = 1/2 - 2
x = -3/2
Therefore, the original fraction is -3/2.
To find the original fraction, let's start by assigning variables.
Let's say the denominator of the fraction is x.
According to the problem statement, the numerator is 2 less than the denominator, so the numerator would be x - 2.
Now let's form the equation based on the given information. When 1½ is added to the original fraction, the result is the reverse of the original fraction.
So, the equation would be:
(x - 2 + 1½) / x = x / (x - 2)
To simplify this equation, we can first clear the fractions by multiplying both sides of the equation by x(x - 2):
x(x - 2) * [(x - 2 + 1½) / x] = x(x - 2) * [x / (x - 2)]
This simplifies to:
(x - 2 + 1½) * (x - 2) = x^2
Now let's expand and solve this equation:
(x - 2 + 1½) * (x - 2) = x^2
(x - 2 + 3/2) * (x - 2) = x^2
(x - 1/2) * (x - 2) = x^2
x^2 - 2x - (1/2)x + 1 = x^2
x^2 - (5/2)x + 1 = x^2
Subtracting x^2 from both sides:
-(5/2)x + 1 = 0
-(5/2)x = -1
x = -1 / -(5/2)
Simplifying:
x = 2/5
Now we have the denominator x = 2/5.
To find the numerator, we can substitute this value back into the equation:
Numerator = x - 2 = (2/5) - 2 = -8/5
However, we were looking for a positive fraction. Since the numerator is negative, there must be a mistake in our calculations.
Please double-check the problem statement or the steps followed to solve the equation.
d - 2 = n ............ (i)
the first sentence (d-2)/d
If 1 1/2 is added it becomes the reversed thus the denominator is 2 less than the numerator
(d - 2)/d + 1 1/2 = (d + 2)/d
Solving the equation d = 3 and n = 1 which is 1/3